Properties

Label 4598.2.a.by
Level $4598$
Weight $2$
Character orbit 4598.a
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{6} + 1) q^{3} + q^{4} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} - 1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{6} + 1) q^{3} + q^{4} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{6} - 1) q^{6} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{7} - q^{8} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{9} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{10} + (\beta_{6} + 1) q^{12} + (\beta_{7} - \beta_{6} + 2) q^{13} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{14} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 2) q^{15} + q^{16} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_1 + 1) q^{17} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1 - 3) q^{18} + q^{19} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{20} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{21} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{23} + ( - \beta_{6} - 1) q^{24} + (2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{25} + ( - \beta_{7} + \beta_{6} - 2) q^{26} + (3 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 5) q^{27} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{28} + (2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1 + 1) q^{29} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{30} + ( - 3 \beta_{7} + \beta_{6} + \beta_{4} - \beta_1 - 1) q^{31} - q^{32} + ( - \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_1 - 1) q^{34} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} - 4) q^{35} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{36} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{37} - q^{38} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - 1) q^{39} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}) q^{40} + (2 \beta_{6} - \beta_{4} - 2 \beta_{2} + 2 \beta_1 - 1) q^{41} + (2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2} + \beta_1 - 1) q^{42} + ( - \beta_{7} + \beta_{6} + \beta_{4} - 3 \beta_1 - 1) q^{43} + (3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 6 \beta_{2} + 5 \beta_1 + 5) q^{45} + (\beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{46} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 - 1) q^{47} + (\beta_{6} + 1) q^{48} + ( - 3 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{49} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{50} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{51} + (\beta_{7} - \beta_{6} + 2) q^{52} + (2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{53} + ( - 3 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 5) q^{54} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{56} + (\beta_{6} + 1) q^{57} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{58} + ( - 3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{59} + (\beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 2) q^{60} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{61} + (3 \beta_{7} - \beta_{6} - \beta_{4} + \beta_1 + 1) q^{62} + ( - 2 \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 6) q^{63} + q^{64} + (2 \beta_{6} + 4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 2) q^{65} + ( - \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{67} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_1 + 1) q^{68} + ( - 2 \beta_{7} - 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 3) q^{69} + (2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} + 4) q^{70} + ( - 3 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{71} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_1 - 3) q^{72} + ( - 3 \beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 1) q^{73} + (\beta_{7} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{74} + (3 \beta_{7} + 3 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 3) q^{75} + q^{76} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + 1) q^{78} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_1 + 2) q^{79} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{80} + (4 \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 6) q^{81} + ( - 2 \beta_{6} + \beta_{4} + 2 \beta_{2} - 2 \beta_1 + 1) q^{82} + ( - 4 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_1 - 4) q^{83} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{84} + ( - \beta_{7} + 5 \beta_{6} - 3 \beta_{4} - 2 \beta_{3} - 3 \beta_1 - 3) q^{85} + (\beta_{7} - \beta_{6} - \beta_{4} + 3 \beta_1 + 1) q^{86} + (5 \beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 8) q^{87} + (2 \beta_{7} - \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 1) q^{89} + ( - 3 \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 5 \beta_1 - 5) q^{90} + ( - 2 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + \beta_{2} + 4 \beta_1 - 3) q^{91} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{92} + ( - 5 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_1 - 1) q^{93} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{94} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2}) q^{95} + ( - \beta_{6} - 1) q^{96} + ( - \beta_{7} + \beta_{6} - 6 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1 + 1) q^{97} + (3 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \beta_1 - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} - 15\nu^{5} + 11\nu^{4} + 64\nu^{3} - 24\nu^{2} - 66\nu - 2 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 20\nu^{3} + 44\nu^{2} - 40\nu - 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + \nu^{6} - 19\nu^{5} - 15\nu^{4} + 108\nu^{3} + 56\nu^{2} - 166\nu - 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 2\nu^{6} - 45\nu^{5} + 14\nu^{4} + 196\nu^{3} + 8\nu^{2} - 218\nu - 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 3\nu^{6} - 87\nu^{5} - 53\nu^{4} + 448\nu^{3} + 232\nu^{2} - 610\nu - 98 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 3\nu^{6} - 109\nu^{5} + 13\nu^{4} + 496\nu^{3} + 72\nu^{2} - 582\nu - 78 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + 2\beta_{5} + \beta_{4} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{7} + \beta_{6} + 8\beta_{5} + 6\beta_{3} + 10\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -25\beta_{7} + 3\beta_{6} + 24\beta_{5} + 8\beta_{4} - 2\beta_{3} + 30\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -83\beta_{7} + 19\beta_{6} + 68\beta_{5} - 4\beta_{4} + 34\beta_{3} + 86\beta_{2} + 124 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -255\beta_{7} + 53\beta_{6} + 236\beta_{5} + 52\beta_{4} - 38\beta_{3} + 8\beta_{2} + 196\beta _1 + 116 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.93433
−0.115899
2.55131
3.02984
−1.84109
1.24609
−0.182716
−2.75320
−1.00000 −2.48823 1.00000 3.25915 2.48823 −4.58972 −1.00000 3.19127 −3.25915
1.2 −1.00000 −2.11129 1.00000 −2.75070 2.11129 −3.66381 −1.00000 1.45756 2.75070
1.3 −1.00000 −0.103249 1.00000 −3.91798 0.103249 4.45623 −1.00000 −2.98934 3.91798
1.4 −1.00000 1.55594 1.00000 2.24033 −1.55594 2.27752 −1.00000 −0.579065 −2.24033
1.5 −1.00000 1.67352 1.00000 −0.566489 −1.67352 −3.40033 −1.00000 −0.199325 0.566489
1.6 −1.00000 2.79123 1.00000 4.02325 −2.79123 −0.274917 −1.00000 4.79095 −4.02325
1.7 −1.00000 3.30349 1.00000 1.88260 −3.30349 −2.41197 −1.00000 7.91304 −1.88260
1.8 −1.00000 3.37860 1.00000 −4.17017 −3.37860 3.60700 −1.00000 8.41492 4.17017
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4598.2.a.by 8
11.b odd 2 1 4598.2.a.cb yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.by 8 1.a even 1 1 trivial
4598.2.a.cb yes 8 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):

\( T_{3}^{8} - 8T_{3}^{7} + 9T_{3}^{6} + 72T_{3}^{5} - 183T_{3}^{4} - 80T_{3}^{3} + 557T_{3}^{2} - 368T_{3} - 44 \) Copy content Toggle raw display
\( T_{5}^{8} - 38T_{5}^{6} + 12T_{5}^{5} + 470T_{5}^{4} - 288T_{5}^{3} - 1984T_{5}^{2} + 1536T_{5} + 1408 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} - 37T_{7}^{6} - 152T_{7}^{5} + 387T_{7}^{4} + 1716T_{7}^{3} - 867T_{7}^{2} - 5408T_{7} - 1388 \) Copy content Toggle raw display
\( T_{13}^{8} - 12T_{13}^{7} + 34T_{13}^{6} + 84T_{13}^{5} - 451T_{13}^{4} + 72T_{13}^{3} + 1088T_{13}^{2} - 96T_{13} - 368 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + 9 T^{6} + 72 T^{5} + \cdots - 44 \) Copy content Toggle raw display
$5$ \( T^{8} - 38 T^{6} + 12 T^{5} + \cdots + 1408 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} - 37 T^{6} + \cdots - 1388 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 12 T^{7} + 34 T^{6} + \cdots - 368 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} - 46 T^{6} + 128 T^{5} + \cdots - 92 \) Copy content Toggle raw display
$19$ \( (T - 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 14 T^{7} - 29 T^{6} + \cdots + 35296 \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} - 123 T^{6} + \cdots + 964 \) Copy content Toggle raw display
$31$ \( T^{8} - 132 T^{6} - 368 T^{5} + \cdots - 9344 \) Copy content Toggle raw display
$37$ \( T^{8} - 24 T^{7} + 134 T^{6} + \cdots + 29569 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} - 130 T^{6} + \cdots - 3872 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} - 96 T^{6} + \cdots - 67328 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} - 172 T^{6} + \cdots - 2659283 \) Copy content Toggle raw display
$53$ \( T^{8} - 36 T^{7} + 413 T^{6} + \cdots + 2770816 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + T^{6} + \cdots - 20032604 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} - 194 T^{6} + \cdots + 16192 \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} - 126 T^{6} + \cdots + 457168 \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{7} - 282 T^{6} + \cdots + 3806464 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} - 270 T^{6} + \cdots - 37382204 \) Copy content Toggle raw display
$79$ \( T^{8} - 12 T^{7} - 78 T^{6} + \cdots + 51808 \) Copy content Toggle raw display
$83$ \( T^{8} + 20 T^{7} - 234 T^{6} + \cdots - 251648 \) Copy content Toggle raw display
$89$ \( T^{8} - 8 T^{7} - 210 T^{6} + \cdots - 320672 \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} - 452 T^{6} + \cdots - 1921664 \) Copy content Toggle raw display
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